In discussions like "FF vs. m43" or "this lens vs that" or "f-number equivalence" or "crop factor", ad nauseam, it is often said that something "gathers more light" but without giving engineering units or any indication how it was figured out.

How is it calculated and what are units of light thus gathered?

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"For every opinion, there is an equal and opposite"TedSD9, SD10, GH1 and a lens or two

If you want a really simplified version...

This image is a part of the simplified object field. It's an infinitely large grid/wall of perfect point lights spaced at an even distance from each other. They all have the same intensity. We're aiming a 2x crop camera with a 50mm lens straight at it, and get this:

2x crop sensor - 50mm lens - f/4.0 aperture

Each individual point included in the image will send light to/through the optical system. How much light from each point that will actually pass through the optical system is determined by the aperture area, or more accurately the front pupil area. Nothing else - at least not in the simplified version.

Let's just say/assume that the aperture used was f/4.0, and that the intensity of the point lights in front of the system each contribute 1 (one) photon to the image with the front pupil area you get at 50mm F4.0. Then you have a total of 150 photons on the image plane (10x15 points are included, 1 photon from each). That was the "total light gathered".

Now - with the same camera - change the lens to a 25mm (half focal length), while keeping aperture at f/4.0. What happens? The field of view widens, and you get four times as many light points included in the image plane.

2x crop sensor - 25mm lens - f/4.0 aperture

-But since the front pupil area is now four times smaller (half focal length, same aperture > four times smaller front pupil area) - you still get the same total amount of light passed through!

Four times as many (150x4 = 600) points of light that each contribute four times less (1/4 = 0.25) photons to the image = still a total of (600x0.25) = 150 photons on the sensor / image plane! Same as with the 50mm F4.0 lens.

That's why the "f/stop" system works - it keeps image plane exposure (light per area on the image plane) constant if you keep the f/# constant - no matter what focal length you choose.Longer focal length > larger front pupil area, but fewer points of light included.Shorter focal length > smaller front pupil area, but more points of light included.

But what happens if we increase the included angle of view by using a 2x larger image plane/sensor (four times larger area) with the 50mm lens in stead? Well, you still include four times as many light points, but the front pupil area doesn't change. So each point still gives the same amount of light on the image plane.

1x sensor - 50mm lens - f/4.0 aperture

So now you have 600 (20x30) points of light included again - just as with the 2x crop sensor and a 25mm lens - but in this case each point still contributes 1 photon to the sensor/image plane - resulting in a total photon count of 600x1.0 = 600 in stead of 600x0.25 = 150.

...................

Then there are some losses, like cos4, mechanical vignetting, optical reflection/absorption and so on - but that's rather beside the point from a purely "simple system" point of view.